High order numerical methods for PDEs on complex domains and their applications in computational biology

有效的数值方法在计算生物学中对偏微分方程模型的研究变得越来越重要。这些时间相关的PDE模型可以是抛物线或双曲线，并定义在复杂的空间域。PI建议在二维（2D）和三维（3D）非结构化网格上对高阶加权基本无振荡（韦诺）格式和间断Galerkin（DG）方法的设计和分析进行研究，用于双曲守恒律方程组和刚性反应扩散方程。PI建议将这些方法应用于发育生物学中形态发生系统的高维模型。了解胚胎发育过程中形态梯度的形成是发育生物学的一个基本问题。PI将研究的数学模型建立在三维复杂几何域上。这些模型的计算分析将揭示BMP形态梯度的形成机制，以及结合胚胎的3D真实形状的重要性。在空间方向上，形态梯度系统通常在胚胎上形成具有3D复杂形状的尖锐梯度。因此，在二维和三维非结构网格上的高阶鲁棒数值方法在处理三维复杂几何区域和解的尖锐梯度的数值模拟中将特别有用。四项具体任务是：1）开发三维四面体网格上的高阶韦诺格式和程序; 2）开发二维和三维非结构网格上的高阶隐式积分因子间断Galerkin（IIF-DG）格式和程序; 3）开发斑马鱼胚胎背腹构图过程中BMP梯度形成的三维反应扩散模型。PI将从基于简化生化基因网络的简单模型出发，以新的生物学实验为动力，建立更完整的模型; 4）通过对三维复杂几何域模型的计算分析，研究形态梯度的形成机制，解决重要的生物学问题。然后，将该模型应用于其他生物，如非洲爪蟾。所提出的数值方法将应用于模拟。这项研究将产生一套很好的建模和计算技术，适用于不同生物胚胎发育过程中各种形态发生系统的定量研究。这些技术有望为形态发生梯度图案形成中复杂现象的计算机模拟做出积极贡献。拟议的活动还将为对数学，计算和生物学接口研究感兴趣的研究生和本科生提供极好的培训和教育机会。学生将通过开发用于计算数学进步的智力和技术工具，并将这些工具应用于生物学中的重要问题，为跨学科研究做好充分准备。

Efficient numerical methods have become more and more important for studying PDE models in computational biology. These time dependent PDE models can be parabolic or hyperbolic, and are defined on complex spatial domains. The PI proposes to perform research in the design, analysis of high order weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin (DG) methods on two-dimensional (2D) and three-dimensional (3D) unstructured meshes, for hyperbolic systems of conservation laws and stiff reaction-diffusion equations. And the PI proposes to apply these methods to high dimensional models for morphogen systems in developmental biology. Understanding morphogen gradient formation during embryo development is a fundamental problem in developmental biology. Mathematical models the PI will study are built on three-dimensional complex geometrical domains. Computational analysis of these models will reveal the formation mechanisms of BMP morphogen gradients, and the importance of incorporating 3D realistic shape of the embryo. In the spatial direction, the morphogen gradient systems usually develop sharp gradients on the embryo with a 3D complex shape. Hence the high order robust numerical methods on 2D and 3D unstructured meshes will be especially useful in the numerical simulations to deal with the 3D complex geometrical domains and the sharp gradients of the solutions. Four specific tasks are: 1) Develop high order WENO schemes and codes on 3D tetrahedral meshes; 2) Develop high order implicit integrating factor discontinuous Galerkin (IIF-DG) schemes and codes on 2D and 3D unstructured meshes; 3) Develop 3D reaction-diffusion models for BMP gradient formation during dorsal-ventral patterning of the zebrafish embryo. Starting from a simple model based on simplified bio-chemical gene network, the PI will create a more complete model motivated by new biological experiments; 4) Via computational analysis of the models on 3D complex geometrical domains, study the formation mechanisms of morphogen gradients, address important biological questions. Then, apply the model to other organisms such as Xenopus. The proposed numerical methods will be applied in the simulations. The proposed research will result in a suite of nice modeling and computational techniques, suitable for quantitative study of various morphogen systems, during the embryo development of different organisms. These techniques are expected to make positive contributions to computer simulations of complicated phenomena in morphogen gradient pattern formation. The proposed activity will also provide excellent training and education opportunities for both graduate and undergraduate students interested in research at the interface of mathematics, computation, and biology. The students will be well prepared to work in interdisciplinary research through developing intellectual and technical tools for the advancement of computational mathematics and applying these tools to important questions in biology.